Problem 47, Proposed by L.E. Dickson, Fellow in Mathematics, University of Chicago:
Prove that (-1)(-1) = +1.
Solutions:
I. Assuming the distributive law to hold, (-1)((+1) + (-1)), or 0, = (-1)(+1) + (-1)(-1). Assuming the commutative law, (-1)(+1) = (+1)(-1 = -1. (-1) + (-1)(-1) = 0, or (-1)(-1) = +1. [L.E. Dickson]
II. (-1)(-1) means that -1 is to be taken subtractively one time. 0 - (-1) = +1. (-1)(-1) = +1. [G.B.M. Zerr]
III. -1 × a = -a, -1 × (a-1) = -(a-1) = -a + 1. -1 × ((a-1)-a) = -a + 1 - (-a) = -a + 1 + a = 1. [M. Philbrick]
...
VII. According to Wood's Elementary Algebra, 17th edition, we have (-5)(-3) = +15. Here -3 is to be subtracted 5 times; that is, -15 is to be subtracted. Now, subtracting -15 is the same as adding +15. Therefore, we have to add +15. Similarly, (-1)(-1) = +1. [W.I. Taylor, F.P. Matz]
VIII. The case (-a)(-b) = +ab is purely conventional and consequently an assumption, which, however, does not deprive the result of its great importance to algebraic operations. [J.F.W. Scheffer]
Amer. Math. Monthly 2 (1895), no.9-10, 272-273

Pasha Zusmanovich

[writings] [soft] [teaching] [seminar] [for students] [talks] [links & files]

Below are given several topics suitable to all levels: PhD, master, bachelor, or even high school. Some of the topics may involve computers, in which cases one should be comfortable in making hands dirty with computer code (in one or more common programming languages and/or computer algebra systems like GAP). The topics are given merely for orientation -- all kind of alterations, variations and additions are possible. See also my writings which usually contain list of questions for further investigations.

For students outside Ostrava and Czech Republic: funded PhD positions are available each spring. Here is a blurb in English and Ukrainian. If interested, please contact me me with informal inquiries.

Various questions related to structure of Lie algebras

Literature:
N. Jacobson, Lie Algebras, Interscience Publ., 1962; reprinted by Dover, 1979.
J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 3rd revised printing, 1980.

In particular:

Other nonassociative algebras

Literature: R.D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, 1966; reprinted by Dover, 1995.

In particular:

Cohomology of Lie algebras

Literature: D.B. Fuchs, Cohomology of Infinite-Dimensional Lie Algebras, Consultants Bureau, N.Y., 1986.
See also a course material on the topic.

Iterative correlation matrices

The correlation coefficient between two sequences measures how "dependent" these sequences are. If we have a number of sequences, representing, say, results of some measurements, their pairwise correlation coefficients can be arranged into correlation matrix. Now one can compute correlation between rows of the correlation matrix itself, getting "correlation matrix of a correlation matrix", and repeat this process. Such procedure is used in applied statistics to get a clustering tree of the initial data. On practice, this iterative procedure "almost always" converges to a pattern consisting of \(-1\) and \(1\), but a rigorous proof of this is lacking, even in the case of \(3 \times 3\) matrices. Some initial matrices exhibit a very strange ("chaotic") behavior. There is a big freedom for computer experimentation.
Literature:
C.-H. Chen, Generalized association plots: information visualization via iterative generated correlation matrices, Statistica Sinica 12 (2002), no.1, 7-29

Craig-Sakamoto

The Craig-Sakamoto theorem says that if \(A\) and \(B\) are \(n \times n\) real symmetric matrices, then \(det(E - aA - bB) = det(E - aA)det(E - bB)\) for any real \(a,b\), iff \(AB = 0\). It has a probabilistic interpretation: if \(X\) is a normally distributed random variable in \(\mathbb R^n\), and \(A\) and \(B\) are \(n \times n\) symmetric matrices, then the random variables \(X^\prime AX\) and \(X^\prime BX\) are independent iff \(AB = 0\). The task would be to generalize the Craig-Sakamoto theorem (for example, by taking more complex polynomial expressions in \(A, B\)), and find probabilistic interpretations of these generalizations.
Literature:
G. Letac, H. Massam, Craig-Sakamoto's theorem for the Wishart distributions on symmetric cones, Ann. Inst. Statist. Math. 47 (1995), no.4, 785-799
J. Ogawa, I. Olkin, A tale of two countries: The Craig-Sakamoto-Matusita theorem, J. Stat. Planning and Inference 138 (2008), no.11, 3419-3428

Homophonic group of Czech language

Consider the free group generated by 26 letters of the English alphabet, and take the quotient by all relations of the form A = B where A and B are two English words spelled differently, but having the same pronunciation. In the article Homophonic quotients of free groups by a group of authors including Don Zagier, it is proved that this group is trivial. The same is true for French. The task is to compute the same group for Czech language. It would be especially interesting if this group will turn out to be nontrivial (which is probable in view of the bigger number of letters in the Czech alphabet, due to diacritics).

Lewis Carroll's soriteses through polynomial systems

Consider the following:
  1. A simpleton, who is not always shouting, is sure to be a crab;
  2. None but spiders are good-humoured;
  3. No unsuccessful frog is despised, so long as it is healthy;
  4. All oysters are good-humoured;
  5. All spiders are healthy, except the green ones;
  6. Unsuccessful crabs, if good-humoured, are popular;
  7. Green crabs are always singing;
  8. The only simpletons, that are popular, are frogs;
  9. Rash young oysters are always unsuccessful;
  10. None but simpletons are good-humoured and yet despised;
  11. No old crabs are healthy;
  12. A rash spider is always despised.
Illustration by Gwynedd M. Hudson, courtesy of British Library.
This is one of the numerous Lewis Carroll's soriteses. Treat this and similar soriteses using modern tools - for example, present it as a system of Boolean polynomial equations, and solve it using the Gröbner bases method (possibly on computer). Compare modern solutions with those proposed by Carroll himself. Do Carroll's methods of solutions of soriteses described in his Symbolic Logic - method of separate syllogisms and method of underscoring - resemble some simple form of words composition in Gröbner bases method?

History of Czech mathematics

Study life and work of some of XIX-early XX century Czech mathematicians. For example: Another topic might be to study history of first Czech mathematical journals. For all this work, you should have a taste for history, and be able to read old mathematical texts in German and French (and occasionally in Czech).

Other history


Created: Mon Jan 19 2015
Last modified: Wed May 22 15:52:53 CEST 2024